The extension consists in considering \((y,w_0,w)\) , where \(w_0\)  may be zero on nondegenerate subintervals of \([0,S_y)\) . On these intervals, the time variable \(t=\int _0^sw_0(\sigma )\,d\sigma \)  is constant –i.e. the time stops–, while the state variable \(y\)  evolves, according to the equation \(y^{\prime }=F(t,y,0,w)\) , sometimes called the fast dynamics. For this reason, system (REF ) is often referred to as the impulsive extension of the original control system (REF ), despite the fact that it is an ordinary control system, as the extended controls \((w_0,w)\)  take values in the compact set \(\overline{\mathbb {U}}\)  and the trajectory \(y\)  is absolutely continuous. A detailed discussion of this topic goes beyond the purposes of the paper. We just mention that an equivalent, \(t\) -based description of this extension, where \(u\)  is no more a function and the trajectory \(x\)  is a discontinuous map whose total variation is bounded on \([0,T]\)  for every \(T<T_x\) , but possibly unbounded on \([0,T_x)\) , in short \(x\in BV_{loc}[0,T_x)\) , could be given (see [1]} and also [2]}, [3]}, [4]}, [5]}).
